Transported addition on the multiplicative natural-number monoid ($a\oplus b={\varphi }^{-1}(\varphi (a)+\varphi (b))$)

For a multiplicative monoid automorphism $\varphi :\mathbb{N}\to \mathbb{N}$ fixing $0$, define $a\oplus b={\varphi }^{-1}(\varphi (a)+\varphi (b))$. This transports a semiring structure to the fixed underlying multiplication, hence a commutative-monoid enrichment on its one-object category. Swapping prime $2$ with an odd prime $p$ gives $1\oplus 1=p$. Infinitely many such primes therefore give infinitely many distinct additions, although the transported enriched categories are isomorphic.